Our second goal here is to circumvent all these limitations. The results in [1]} are again based on a Bernstein method. Apart from the issues on the generality of the statements in [1]}, a crux is that the Bernstein approach may not be employed for different problems, e.g. involving degenerate or fractional diffusions. Therefore, a general goal of this work is to develop a different approach to maximal regularity, which can be flexible enough to be applied to a wider range of equations (which will be briefly described at the end of this introduction). The core idea of the method proposed here is to get maximal regularity using Hölder estimates as an intermediate step. Nevertheless, (linear) maximal regularity will be crucial to obtain such Hölder estimates, so one may argue, after reading the proofs, that regularity at these two scales is in fact interconnected.
